Measurable Riemannian geometry on the Sierpinski gasket: the Kusuoka measure and the Gaussian heat kernel estimate

We study the standard Dirichlet form and its energy measure,called the Kusuoka measure, on the Sierpinski gasket as aprototype of “measurable Riemannian geometry”. The shortest pathmetric on the harmonic Sierpinski gasket is shown to be thegeodesic distance associated with the “measurable Riemannianstructure”. The Kusuoka measure is shown to have the volumedoubling property with respect to the Euclidean distance and alsoto the geodesic distance. Li–Yau type Gaussian off-diagonal heatkernel estimate is established for the heat kernel associated withthe Kusuoka measure.     

Singularly perturbed elliptic systems

We study the existence and concentration behavior of positive solutions for a class of Hamiltonian systems (two coupled nonlinear stationary Schrödinger equations). Combining the Legendre–Fenchel transformation with mountain pass theorem, we prove the existence of a family of positive solutions concentrating at a point in the limit, where related functionals realize their minimum energy. In some cases, the location of the concentration point is given explicitly in terms of the potential functions of the stationary Schrödinger equations.     

Another approach to the heat flow for Plateau problem

For the minimal surfaces in Rⁿ with Plateau boundary condition and establish the global existence and uniqueness of the flow as well as the continuous dependence of the initial datum.     

Heat kernel expansions in vector bundles

Let M be a complete connected smooth (compact) Riemannian manifold of dimension n. Let Π:V→M be a smooth vector bundle over M. Let be a second order differential operator on M, where Δ is a Laplace-Type operator on the sections of the vector bundle V and b a smooth vector field on M. Let k_t(−,−) be the heat kernel of V relative to L. In this paper we will derive an exact and an asymptotic expansion for k_t(x,y₀) where y₀ is the center of normal coordinates defined on M, x is a point in the normal neighborhood centered at y₀. The leading coefficients of the expansion are then computed at x=y₀ in terms of the linear and quadratic Riemannian curvature invariants of the Riemannian manifold M, of the vector bundle V, and of the vector bundle section ? and its derivatives.We end by comparing our results with those of previous authors (I. Avramidi, P. Gilkey, and McKean-Singer).     

Existence and concentration of positive solutions for a class of gradient systems

Some gradient systems with two competing potential functions are considered. Bound states (solutions with finite energy) are proved to exist and to concentrate at a point in the limit. The proof relies on variational methods, where the existence and concentration of positive solutions are related to a suitable ground energy function.     

Partial differential equations with differential constraints

A geometric setting for constrained exterior differential systems on fibered manifolds with n-dimensional bases is proposed. Constraints given as submanifolds of jet bundles (locally defined by systems of first-order partial differential equations) are shown to carry a natural geometric structure, called the canonical distribution. Systems of second-order partial differential equations subjected to differential constraints are modeled as exterior differential systems defined on constraint submanifolds. As an important particular case, Lagrangian systems subjected to first-order differential constraints are considered. Different kinds of constraints are introduced and investigated (Lagrangian constraints, constraints adapted to the fibered structure, constraints arising from a (co)distribution, semi-holonomic constraints, holonomic constraints).     

Le problème de Yamabe avec singularités

Let (Mn,g) be a compact riemannian manifold of dimension n?3. Under some assumptions, we prove that there exists a positive function φ solution of the Yamabe equation

     

Periodic solutions for an asymptotically linear wave equation with resonance

In this paper, we study the existence and multiplicity of nontrivial periodic solutions for an asymptotically linear wave equation with resonance, both at infinity and at zero. The main features are using Morse theory for the strongly indefinite functional and the precise computation of critical groups under conditions which are more general.     

Construction of the fundamental solution of disturbed parabolic equation

In present paper the parabolic equation solution is built. The construction is reduced to iterative procedure. And convergence of the latter is proven.     

Asymptotics of the porous media equation via Sobolev inequalities

Let M be a compact Riemannian manifold without boundary. Consider the porous media equation , u(0)=u₀∈L^q, ? being the Laplace-Beltrami operator. Then, if q?2∨(m-1), the associated evolution is L^q-L^∞ regularizing at any time t>0 and the bound ‖u(t)‖_∞?C(u₀)/t^β holds for t<1 for suitable explicit C(u₀),γ. For large t it is shown that, for general initial data, u(t) approaches its time-independent mean with quantitative bounds on the rate of convergence. Similar bounds are valid when the manifold is not compact, but u(t) approaches u≡0 with different asymptotics. The case of manifolds with boundary and homogeneous Dirichlet, or Neumann, boundary conditions, is treated as well. The proof stems from a new connection between logarithmic Sobolev inequalities and the contractivity properties of the nonlinear evolutions considered, and is therefore applicable to a more abstract setting.     

Positive solutions of semilinear biharmonic equations with critical Sobolev exponents

In this paper we study the critical growth biharmonic problem with a parameter λ and establish uniform lower bounds for Λ, which is the supremum of the set of λ, related to the existence of positive solutions of the biharmonic problem.     

Inverse problems and index formulae for Dirac operators

We consider a Dirac-type operator DP on a vector bundle V over a compact Riemannian manifold (M,g) with a non-empty boundary. The operator DP is specified by a boundary condition P(u_|∂M)=0 where P is a projector which may be a non-local, i.e., a pseudodifferential operator. We assume the existence of a chirality operator which decomposes L²(M,V) into two orthogonal subspaces X₊⊕X₋. Under certain conditions, the operator DP restricted to X₊ and X₋ defines a pair of Fredholm operators which maps X₊→X₋ and X₋→X₊ correspondingly, giving rise to a superstructure on V. In this paper we consider the questions of determining the index of DP and the reconstruction of and DP from the boundary data on ∂M. The data used is either the Cauchy data, i.e., the restrictions to ∂M×R₊ of the solutions to the hyperbolic Dirac equation, or the boundary spectral data, i.e., the set of the eigenvalues and the boundary values of the eigenfunctions of DP. We obtain formulae for the index and prove uniqueness results for the inverse boundary value problems. We apply the obtained results to the classical Dirac-type operator in M×C⁴, M⊂R³.     

On the existence of pair positive–negative solutions for resonance problems

This paper deals with a class of semilinear elliptic Dirichlet boundary value problems at resonance. We introduce a sufficient Landesman–Lazer condition for the existence of pair positive–negative solutions. Furthermore, developing the fibering method in the framework of the Leray–Schauder degree theory we can prove the existence of branches for positive and negative solutions.     

On some degenerate quasilinear equations involving variable exponents

In the present paper, we are concerned with some degenerate quasilinear equations involving variable exponents. Using various (variational and nonvariational) techniques, we prove existence, nonexistence and multiplicity results.     

Stability of symmetric vortices for two-component Ginzburg–Landau systems

We study Ginzburg–Landau equations for a complex vector order parameter Ψ=(_ψ+,_ψ−)∈C²$\Psi =({\psi }_{+},{\psi }_{-})\in {\mathbb{C}}^2$. We consider the Dirichlet problem in the disk in R²${\mathbb{R}}^2$ with a symmetric, degree-one boundary condition, and study its stability, in the sense of the spectrum of the second variation of the energy. We find that the stability of the degree-one equivariant solution depends on the Ginzburg–Landau parameter as well as the sign of the interaction term in the energy.     

Harmonic maps from manifolds of <Emphasis Type="Italic">L</Emphasis><Superscript>∞</Superscript>-Riemannian metrics

For a bounded domain Ω ⊂ R ⁿ endowed with L ^∞-metric g, and a C ⁵-Riemannian manifold (N, h) ⊂ R ^k without boundary, let u ∈ W ^1,2(Ω, N) be a weakly harmonic map, we prove that (1) u ∈ C ^α (Ω, N) for n = 2, and (2) for n ≥ 3, if, in additions, g ∈ VMO(Ω) and u satisfies the quasi-monotonicity inequality (1.5), then there exists a closed set Σ ⊂ Ω, with H ^n-2(Σ) = 0, such that for some α ∈ (0, 1). C. Y. Wang Partially supported by NSF.     

Symmetric vortices for two-component Ginzburg–Landau systems

We study Ginzburg–Landau equations for a complex vector order parameter Ψ=(_ψ+,_ψ−)∈C²$\Psi =({\psi }_{+},{\psi }_{-})\in {\mathbb{C}}^2$. We consider symmetric vortex solutions in the plane R²${\mathbb{R}}^2$, ψ(x)=_f±(r)e^i_n±θ$\psi (x)=f_{\pm }(r)e^{i{n}_{\pm }\theta }$, with given degrees _n±∈Z$n_{\pm }\in \mathbb{Z}$, and prove the existence, uniqueness, and asymptotic behavior of solutions as r→∞$r\to \infty$. We also consider the monotonicity properties of solutions, and exhibit parameter ranges in which both vortex profiles _f+$f_{+}$, _f−$f_{-}$ are monotone, as well as parameter regimes where one component is non-monotone. The qualitative results are obtained by means of a sub- and super-solution construction and a comparison theorem for elliptic systems.     

A nondifferentiable extension of a theorem of Pucci and Serrin and applications

We study the multiplicity of critical points for functionals which are only differentiable along some directions. We extend to this class of functionals the three critical point theorem of Pucci and Serrin and we apply it to a one-parameter family of functionals Jλ, λ∈I⊂R. Under suitable assumptions, we locate an open subinterval of values λ in I for which Jλ possesses at least three critical points. Applications to quasilinear boundary value problems are also given.